Your model violates the property that if S(m) = S(n), then m=n, because S(1) = S(0) yet 1 != 0. You might try to patch this by changing the model so it only has 0 as an element, but there is a further axiom that says that 0 is not the successor of any number.

Together, the two axioms used above can be used to show that the natural numbers 0, S(0), S(S(0)), etc. are all distinct. The axiom of induction can be used to show that these are all the natural numbers, so that we can't have some extra "floating" integer x such that S(x) = x.